NBS

355

Correlations for Predicting Leakage

Through Closed Valves

JESSE HORD

Q

Z

s

1*1 0F

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^CAU Of

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NBS TECHNICAL NOTE 355

ISSUED AUGUST 1, 1967

CORRELATIONS FOR PREDICTING LEAKAGE

THROUGH CLOSED VALVES

JESSE HORD

Cryogenics Division

Institute for Materials Research

National Bureau of Standards

Boulder, Colorado

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TABLE OF CONTENTS

Page

ABSTRACT 1

1 . Introduction 2

2. Correlation of Flow Formulae 3

2. 1 Free Molecule Flow Through Orifices and

Short Channels or Through Long Channels (Kn - 1) • 4

2.2 Continuum Flow Through Orifices (Kn - 0.01) • • • 5

2.3 Viscous (Poiseuille) Flow Through Long Channels

(Kn ^0.01) 5

2.4 Turbulent, Incompressible Flow Through Long

Channels (Kn ^ 0.01) 6

2. 5 Transition Flow Through Orifices and Short Tubes

(0.01 < Kn < 1.0) 7

2. 6 Transition Flow Through Long Channels

(0.01 < Kn < 1.0) 7

2.7 Compressible Flow Through Long Channels

(Kn ^ 0.01) 8

2.8 Critical Flow Through Orifices and Long Channels

(Kn £ 0.01) 8

2.9 Discussion of the Analytical Results 9

3. Summary 10

Table 3. 1 Summary of Formulae 12

4. Nomenclature 13

5. References 14

in

Correlations for Predicting Leakage Through Closed Valves

Jesse Hord

ABSTRACT

Safety, convenience, and economy often demand the inference of

leakage of a given fluid from known leakage of another test fluid across

valve seats, welded and threaded fittings, seals, etc. The temperatures

of the fluids also may be different. An example is the prediction of

gaseous hydrogen leakage at 20°K from test data obtained with nitrogen

gas at 77°K. Various flow formulae (molecular, transition, and continuum) are examined, and two simple methods of correlating leakage for

single-phase fluids are deduced. The correlations obtained (excluding

transition flow through long channels) indicate the leakage is inversely

proportional to the square root of the density, or inversely proportional

to the absolute viscosity, of the fluid. Thus, for gases, the leakage is

directly proportional to the sonic velocity of the gas. The proper relationship must be established by experiment for each valve, fitting, etc.

Key Words: Leakage through valves, leak rates, leak detection,

predicting leak rates, valve seat leakage, leak flow

rates, seal leakage.
1. Introduction

A large volume of literature has been devoted to detection and

measurement of leakage flow and to the writing of leak test specifications.

The problems involved in establishing meaningful specifications have

been outlined by Hutchins[ 1] and Baumann[ 2] . Various flow theories have

been utilized in some recent articles[ 2-10] to estimate leakage for specific

problems; however, it appears there are no general guidelines for the

inference of leakage of a given fluid from the known leakage of another

test fluid. It is often desirable to leak test equipment with a non-hazardous

fluid (at some convenient temperature) and then infer the leakage of a

hazardous fluid (perhaps at a less convenient temperature). It is the purpose of this paper to provide these leakage correlations. There are manyvariables involved in a problem of this sort, and thus many assumptions

are required to obtain useful engineering results. Although we are primarily concerned with valve seat leakage, the results are applicable to

any leakage path (e. g. threaded and welded fittings, seals, etc. ). Thermal

contraction and expansion of materials may alter the leak-path dimensions

and make it quite difficult to correlate leakage results from one temperature to another. For valves, the seat seal resilience may change with

temperature, and also the open-close cycle history affects the leakage as

the seal wears in or wears out. Thus, it is obvious that correlating

leakage from fluid-to -fluid and from one temperature to another is risky,

and testing of the item in the actual use -environment is recommended;

however, where correlation techniques must be used, the results of the

following analysis are offered as guidelines. The flow formulae are given

in volumetric form since leakage is normally quoted in volumetric units.
2. Correlation of Flow Formulae

In order to obtain useful engineering results, it is necessary to

make several simplifying assumptions concerning the leakage flow:

a) The flow is isothermal for orifice flow in the molecular regime

and for pipe flow in all regimes; this is normally a good assumption

for relatively large-mass equipment with very little leakage.

b) Where continuum orifice flow applies, the leak-path cross

section is considered small enough to neglect variations in velocity

of approach, and the Reynolds number is large enough to assure a

nearly constant discharge coefficient for all fluids at all inlet temperatures considered.

c) The pressure drop and absolute pressure levels across the

leakage path are identical at all temperatures for all fluids. This

assumption eliminates change in leak-path geometry with pressure.

d) The leak -path geometry does not change with temperature. For

those cases where the geometry change may be computed with

confidence, an area correction may be applied to the correlation.

e) The flow velocity does not exceed the velocity of sound in the

fluid at specified conditions (i. e. adiabatic or isothermal flow, etc. ).

f ) For all gases of concern, the ratio of specific heats and thus the

critical (sonic flow) pressure ratio for orifices and nozzles do not

differ appreciably. This is a good assumption for most gases over

a wide range of temperatures at fairly low pressures. Variations

in specific heat ratios cause relatively small changes[ 11] in the

expansion factor, Y. The assumption of relatively small variations

in specific heat ratios also permits isothermal and adiabatic acoustic
velocity ratios, to be interchanged. Isentropic conditions are

assumed where critical flow occurs in orifices with contoured

inlets

.

g) For the case of viscous (Poiseuille) flow, the frictional resistance is assumed to vary from fluid to fluid; the friction factor

is taken as constant in the turbulent flow formulae. Assumption

of a constant friction requires either rough flow channels or relatively large Reynolds numbers, both of which appear tenable.

Additional assumptions which pertain to individual flow processes are

noted in that which follows.

2. 1 Free Molecule Flow Through Orifices and

Short Channels or Through Long Channels (Kn ^ 1)

Knudsen flow applies when the gas mean free path is about equivalent to the smallest characteristic dimension of the flow duct (i.e. Kn?al),

The volumetric flow rate may be written in the form

Q = C (AP/P ) (RT)

1/2

(2.1a)

and the leakage Q of fluid A at temperature A may be obtained from a known leakage Q of fluid B at temperature B via

B

Q A /Gl = (R A T /R T ) A B A a B B

1/2

(2.1b)

The constant in (2. la) includes flow geometry parameters and consequently is different for long channels, short channels, and orifices, but

the correlation result given by (2. lb) is the same (since the flow path

geometry is assumed to remain unchanged with variations in temperature)
2.2 Continuum Flow Through Orifices (Kn ^ 0.01)

The volumetric flow rate for both compressible and incompressible

fluids is given by

Q = C

2

Y(AP/p

i

) 1/2

, (2.2a)

and the correlating ratio is

1/2 QA

/QB

=(P 1B /P 1A> • < 2 - 2b >

In (2.2a) C involves the discharge coefficient and leak-path geometry,

and Y is the expansion factor, all of which are invariant under the foregoing assumptions.

2.3 Viscous (Poiseuille) Flow Through Long Channels (Kn ^ 0.01)

The flow rate given here is subject to the restrictions of Poiseuille

flow; i.e. , incompressible, fully-developed laminar flow, through long,

narrow channels. This formulation is valid for laminar, isothermal compressible flow where the Mach number is less than one-third [12].

Q = C AP/n , (2.3a)

and the correlation factor is

QA

/QB=V^A' < 2 ' 3b >

Weiner [5] has presented some leakage flow nomographs based on Poiseuille

flow relations and suggested basic criteria and definitions for fluid leakage.

Some care must be taken in computing liquid leakage since surface tension

can influence the results [5,8] when the leakage gap and flow rates are very

small.
2.4 Turbulent, Incompressible Flow

Through Long Channels (Kn ^ 0.01)

It is convenient to use constant frictional resistance to obtain

Q = C

4

(AP/fp)

1/2

, (2.4a)

and

VQ

B = (P B

/P A

)1/2

- < 2 - 4b »

The mean density is used to account for gas flow where it may be considered incompressible.

Steward [13] has suggested the friction factor be expressed as a function of Reynolds number and treated as a variable in (2. 4a) to obtain

a correlating ratio which accounts for viscosities in addition to densities,

Using the relationship f =a/Re and performing the indicated algebra

there results

P. q*

/q b -tfj

2 " p [^

2 "

Examination of (2. 4c) indicates that the viscosity function is fairly weak

for practical values of |3(~ to 0.2), and the density function does not differ

significantly from that of (2. 4b). In consideration of the assumptions

applied in the treatment of the various formulae in this paper, the refinement offered by (2. 4c) is probably not warranted; however, it may have

some limited application in liquid-to-gas or gas -to-liquid correlations

where the test fluids are considered relatively incompressible and no

phase change occurs across the leakage path. Equation (2.4c) is offered

for reference only and is not applied to the general conclusions of this
paper. Note that (2. 4c) reduces to (2. 3b) for laminar flow (3=1) and is

equivalent to (2 . 4b) for constant friction (|3 =0). Use of (2. 4c) requires

experimental determination of (3.

2. 5 Transition Flow Through Orifices and Short Tubes

(0.01 < Kn < 1.0)

Transition flow is intermediate to free molecule and continuum

flow, and the theory for prediction of flow through orifices and short tubes

in this region is not complete. Interpolation between free molecule and

continuum formulae is common [14] for orifices; however, both the continuum and free molecule flow are inversely proportional to the square

root of the density of the fluid, and thus produce identical correlating

ratios (the equivalency of (2. lb) and (2.2b) is shown later). Some work

on transition flow through orifices [15] and short tubes [16] was recently

reported. Sreekanth [15] was able to correlate his experimental data for

orifices with the theory for transition flow through long tubes by using a simple geometrical correction factor. Correlating ratios obtained from

his formula would be identical to those for transition flow through long tubes

2 . 6 Transition Flow Through Long Channels (0.01 < Kn < 1 . 0)

Transition flow through long channels is adequately represented by

the summation of Knudsen and Poiseuille flow,

Q = [(Cj/P) (RT)

1/2

+ C /|i] AP. (2.6a)

Correlation of (2. 6a) is slightly cumbersome and is obviously a combination of density and viscosity effects. This formula has been treated in the

literature and will not be pursued further here. Guthrie and Wake rling [17]

have capably handled the problem and tabulated data for several gases.

Graham [10] has also expanded somewhat on this particular subject and
provided detailed discussion on leak-rate specifications and testing. In

addition, he discusses conversion of leakage rates to standard conditions.

A bibliography and review of various leakage analyses has been prepared

by Bauer, et al. [18]. Nomographs for incompressible and compressible

viscous flow and transition flow through a circular interface are given [18].

2.7 Compressible Flow Through Long Channels (Kn ^ 0.01)

In addition to the foregoing formulae, we may write expressions

for the flow of compressible fluids in long channels for i) isothermal flow,

laminar or turbulent, with friction and ii) adiabatic flow with or without

friction. In general, these equations are too complex for the cross-correlation desired here. As stated in the assumptions, the isothermal

case is probably most prevalent. For the case of isothermal compressible

flow with constant friction, the volume flow rate is

Q = C

AP/P 1/2

5 \^ln (Pj/^) + F

(2.7a)

and the correlation is identical to (2.4b). Although the results would be

rather cumbersome, the friction parameter in (2. 7a) could be considered

variable as in the development of (2.4c).

2.8 Critical Flow Through Orifices and Long Channels (Kn ^ 0.0.1)

When the downstream pressure is less than about 1/2 the upstream

pressure, sonic flow may occur through nozzles or orifices with rounded

inlets. It is usually very difficult to determine if critical flow exists in

long channels, and the critical pressure ratio varies considerably. For

flow at acoustic velocity, the volumetric flow rate in both orifices and

pipes may be expressed by an equation of the form

Q = C.(RT)

D 1/2

(2.8a)
and is correlated by (Z. lb). Vance [19] obtained an equivalent relationship

in discussing the performance of throttling valves where sonic flow is

attained; however, he states that "the volume flow ratio is proportional to

the ratio of sonic velocities at inlet conditions and is independent of the

gas or temperature." Obviously, this statement is erroneous.

Z. 9 Discussion of the Analytical Results

In considering assumption (c) we note that the mean density P and

the upstream density P are linearly related and consequently (Z.Zb) and

(Z.4b) produce identical results. Use of the ideal gas law shows (Z. lb)

is also identical to (Z.Zb) and (Z.4b). Two very simple relationships are

then deduced for use as leakage correlating factors from fluid-to -fluid

(i. e. , from one fluid at one temperature to another fluid at another temperature). The leakage rates are inversely proportional to the absolute

viscosity or inversely proportional to the square root of the density of

the respective fluids. For gases, the leakage rates are directly proportional to the sonic velocity in the gas. Perhaps the most useful way of

presenting the results is

QA/QB

=< P 1B /P 1A

)1/Z

- (2 ' 9a)

or

VQB=V*A- (2 ' 9b)

It is recommended that experimental data be taken with two different

fluids to establish the proper correlating procedure, i.e., (Z.9a) or(Z.9b)

prior to inferring the leakage at the desired use-temperature. It is

possible that in many cases the leakage may be bracketed by (Z.9a) and
(2. 9b), and the more pessimistic relation may be used. Equation (2. 9a)

is valid only for liquid-to-liquid or gas-to-gas correlations while (2. 9b)

may also be used for liquid-to-gas or gas-to-liquid correlations (within

the framework of Poiseuille flow). Equation (2. 9a) is limited to similarphase correlations because the friction factor changes with fluid phase,

and the compressible and incompressible formulae differ for both pipe

and orifice flow. The correlating ratios developed are related to the

volumetric flow of the individual fluids at their respective inlet temperatures. If it is desired to relate the leakage ratio to some reference temperature and pressure (e. g. , STP), (2. 9a) and (2. 9b) must be multiplied

by the appropriate density ratios. For gas-to-gas correlations at STP,

(2. 9a) and (2. 9b) should be multiplied by T /T , and the results may be

conveniently written

or

(QA

/QB>STP - [ <VT

A><

M

B /MA

)]1/2

' < 2 ' 9C)

(QV /Q ) = (M- /V> ) (T /T ). A

WB

; STP l B' A' l B A'

(2.9d)

3. Summary

Two very simple relations are deduced to aid in the inference of

leakage through closed valves, etc. The leakage of fluid A at temperature A is obtained from the known leakage of fluid B at temperature B,

and the results are summarized in table 3.1. The proper relationship must

be established for each valve (or item) by testing with at least two different fluids, preferably at different temperatures. The two equations may

be used to bracket the leakage and the more pessimistic formula used for

inference of leakage at the use-temperature.

10
Equations (2. 9a) and (2. 9b) are recommended only when actual use- environment tests cannot be performed. Equation (2.4c) is applicable to

liquid-to-gas or gas-to-liquid correlations for those particular cases

where single-phase, incompressible flow can be assumed. Transition

flow through long channels was not included in this summary since it has

been capably treated in the literature[ 10, 17, 18] . None of the formulae

presented here pertain to two-phase or mixed flow.

11
Table 3. 1 Summary of Formulae

Type of Flow Applications & Restrictions Correlating

Ratio

Free Molecule Flow

through orifices and

short channels or

through long channels

Subsonic continuum Equation (2.9a) is limited to

flow through orifices liquid -to -liquid or gas-to-gas

computations

Turbulent, incom- Eq. (2.9a)

pressible flow through

long channels with

constant friction

-'-Transition flow ^Better results may be obthrough orifices tained by treating this case

Isothermal compressible flow through

as transition flow through

long tubes [15].

long channels with

constant friction

Critical flow through

rounded -orifices

,

nozzles, and long

channels

Viscous (Poiseuille) Equation (2. 9b) may be used Eq. (2.9b)

flow through long for gas-to-gas, liquid-tochannels liquid, liquid-to-gas, or gasto-liquid correlations as long as the requirements of Poiseuille flow are satisfied

Transition flow See References [10, 17, 18]

through long channels Applies to gases only.

12
4. Nomenclature

C = Constant in flow formulae

F = Frictional resistance constant

f = Friction factor (= a/Re ) Kn = Knudsen number (- ratio of the mean-free

path of the gas flowing through the duct to

the smallest characteristic dimension of

the duct), applied only to gas flow

M = Molecular weight

P = Upstream or inlet absolute pressure

P = Downstream or exit absolute pressure

P = Arithmetic mean absolute pressure

(^ [P

2 + P

2

]/2)

AP = P - P

1 2

Q = Volumetric flow rate

R = Gas constant, equals universal gas constant

divided by the molecular weight

Re = Reynolds number (based on flow area

dimensions)

STP = Denotes standard temperature and pressure

T = Absolute temperature of fluid at inlet conditions

Y = Expansion factor in orifice formulae

Greek

a = Constant in friction factor formula

P = Constant in friction factor formula

p = Upstream or inlet fluid density

p = Downstream or exit fluid density

p = Arithmetic mean fluid density (= [P + P ]/2)

(J, = Absolute viscosity of the fluid at inlet conditions

13
5. References

1. Hutchins, W. C, "Leak Test Specifications," Instruments and

Control Systems, Vol. 35, p. 107-109 (April 1962).

2. Baumann, H. D. , "Should a Control Valve Leak?" Instruments

and Control Systems, Vol. 39, p. 1 1 1 -1 15 (August 1966).

3. Gunkel, V. K. , "Gas Filled Electronic Enclosures, " Journal of

Environmental Sciences, Vol. 9, p. 25-28 (August 1966).

4. McHugh, J. D. , "Adiabatic, Laminar Flow in Concentric, Sleeve

Seals, " Lubr. Engr. , Vol. 22, p. 17-22 (January 1966).

5. Weiner, R. S. , "Basic Criteria and Definitions for Zero Fluid

Leakage," JPL Tech. Rep't No. 32-926 (December 1966).

6. Snapp, R. B. , "Theoretical Analysis of Face-Type Seals with

Varying Radial Face Profiles, " ASME Paper 64-WA/LUB-6

(December 1964).

7. McGrew, J. M. , and J. D. McHugh, "Analysis and Test of the

Screw Seal in Laminar and Turbulent Operation, " ASME Journ.

Basic Engr., Vol. 87, p. 153-162 (March 1965).

8. Bauer, P. , "Investigations of Leaking and Sealing Parameters, "

Tech. Report AFRPL TR-65-153 consisting of 678 pages

(August 1965).

9. Vermes, G. , "A Fluid Mechanics Approach to the Labyrinth Seal

Leakage Problem, " ASME Journ. for Engr. Power, Vol. 83,

p. 161-169 (April 1961).

10. Graham, J. W. , "Gas Leakage in Sealed Systems," Chem.

Engineering, p. 169 -1 74 (May 1 1

, 1964).

11. "Flow Measurement, " Chap. 4, Part 5, Measurement of Quantity

of Materials, PTC 19. 5; 4-1959, ASME 29 West 39th Street,

New York, New York, p. 68-78.

14
12. Schlichting, H. , Boundary Layer Theory, McGraw Hill Book Co.,

Inc., New York, N. Y. , p. 10(1955).

13. Steward, W. G. , private communication.

14. Guthrie, A. , and R. K. Wakerling, Vacuum Equipment and Tech -

niques, First Ed., McGraw Hill Book Co., Inc., New York, New

York, p. 20 (1949).

15. Sreekanth, A. K. , "Transition Flow Through Short Circular Tubes,"

Phys. of Fluids, Vol. 8, p. 1951 -1956 (November 1965).

16. Carley, C. T. , Jr., and F. O. Smetana, "Experiments on Transition Regime Flow Through a Short Tube 'with a Bellmouth Entry, " AIAA Journ. , Vol. 4, No. 1, p. 47-54 (January 1966).

17. Ibid reference 14, p. 190.

18. Bauer, P., M. Glickman, and F. Iwatsuki, "Dynamic Seals,"

Tech Rep't AFRPL TR 65-61, Vol. II consisting of 260 pages

(May 1965).

19. Vance, R. W. , Applied Cryogenic Engineering, John Wiley and

Sons, Inc., New York, New York, p. 25 (1962).

15

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